Assume that \(a\) and \(b\) are real numbers, and the two zero points of the quadratic polynomial \(\lambda^2- a\lambda-b\) are \(\lambda_1=r\) and \(\lambda_2=s\). J. Matkowski and W. Zhang [Acta Math. Sin., New Ser. 13, No. 3, 421-432 (1997; Zbl 0881.39011)] obtained general solutions of the iterated functional equation \[ f^2(x)=af(x)+bx,\text{ normalsize for all } x\in R; \quad f\in C^0(R, R)\tag{1} \] for the three cases \(0<r<s\), \(r<0<s\neq -r\), and \(r=s\neq 0\) and proved that there are no solutions of equation (1). M. Kuczma [Functional equations in a single variable (1968; Zbl 0196.16403)] found the general solution for (1). In this paper, the general solution of (1) is given for the remaining two cases \(r<s<0\) and \(rs=0\). In addition a simple proof about the general solution of (1) is given for \(r<0<s\neq-r\).